# Otten, Gilbertson, Males, & Clark (2014)

The Mathematical Nature of Reasoning-and-Proving Opportunities in Geometry Textbooks

## Abstract

International calls have been made for reasoning-and-proving to permeate school mathematics. It is important that efforts to heed this call are grounded in an understanding of the opportunities to reason-and-prove that already exist, especially in secondary-level geometry where reasoning-and-proving opportunities are prevalent but not thoroughly studied. This analysis of six secondary-level geometry textbooks, like studies of other textbooks, characterizes the justifications given in the exposition and the reasoning-and-proving activities expected of students in the exercises. Furthermore, this study considers whether the mathematical statements included in the reasoning-and-proving opportunities are general or particular in nature. Findings include the fact that the majority of expository mathematical statements were general, whereas reasoning-and-proving exercises tended to involve particular mathematical statements. Although reasoning-and-proving opportunities were relatively numerous, it remained rare for the reasoning-and-proving process itself to be an explicit object of reflection. Relationships between these findings and the necessity principle of pedagogy are discussed.

• Background
• Research on Students' Reasoning-and-Proving
• Theoretical Perspective
• Method
• Sample
• Analytic Framework
• Characterizing Mathematical Statements
• Separating Mathematical Statements from Justification
• Separating Opportunities to Prove from Opportunities to Explain
• Codes Inherited from Thompson, Senk, and Johnson
• Codes Added Based on Pilot Analysis of Geometry Textbooks
• Statements or Exercises About Reasoning-and-Proving
• Analytic Procedures
• Results
• Reasoning-and-Proving in Textbook Exposition
• Types of Statements in Textbook Exposition
• Types of Justifications in Textbook Exposition
• Summary
• Reasoning-and-Proving in Student Exercises
• Types of Statements in Student Exercises
• Types of Reasoning-and-Proving Activities in Student Exercises
• Statement-Types of Proof-Focused Exercises
• Summary
• Comparing Textbook Exposition to Student Exercises
• Discussion

## Also

APA
Otten, S., Gilbertson, N. J., Males, L. M., & Clark, D. L. (2014). The mathematical nature of reasoning-and-proving opportunities in geometry textbooks. Mathematical Thinking and Learning, 16(1), 51–79. doi:10.1080/10986065.2014.857802
BibTeX
@article{Otten2014,
abstract = {International calls have been made for reasoning-and-proving to permeate school mathematics. It is important that efforts to heed this call are grounded in an understanding of the opportunities to reason-and-prove that already exist, especially in secondary-level geometry where reasoning-and-proving opportunities are prevalent but not thoroughly studied. This analysis of six secondary-level geometry textbooks, like studies of other textbooks, characterizes the justifications given in the exposition and the reasoning-and-proving activities expected of students in the exercises. Furthermore, this study considers whether the mathematical statements included in the reasoning-and-proving opportunities are general or particular in nature. Findings include the fact that the majority of expository mathematical statements were general, whereas reasoning-and-proving exercises tended to involve particular mathematical statements. Although reasoning-and-proving opportunities were relatively numerous, it remained rare for the reasoning-and-proving process itself to be an explicit object of reflection. Relationships between these findings and the necessity principle of pedagogy are discussed.},
author = {Otten, Samuel and Gilbertson, Nicholas J. and Males, Lorraine M. and Clark, D. Lee},
doi = {10.1080/10986065.2014.857802},
journal = {Mathematical Thinking and Learning},
number = {1},
pages = {51--79},
title = {{The mathematical nature of reasoning-and-proving opportunities in geometry textbooks}},
url = {http://www.tandfonline.com/doi/abs/10.1080/10986065.2014.857802},
volume = {16},
year = {2014}
}